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Theorem ordtri3 4609
Description: A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtri3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )

Proof of Theorem ordtri3
StepHypRef Expression
1 ordirr 4591 . . . . . 6  |-  ( Ord 
A  ->  -.  A  e.  A )
2 eleq2 2496 . . . . . . 7  |-  ( A  =  B  ->  ( A  e.  A  <->  A  e.  B ) )
32notbid 286 . . . . . 6  |-  ( A  =  B  ->  ( -.  A  e.  A  <->  -.  A  e.  B ) )
41, 3syl5ib 211 . . . . 5  |-  ( A  =  B  ->  ( Ord  A  ->  -.  A  e.  B ) )
5 ordirr 4591 . . . . . 6  |-  ( Ord 
B  ->  -.  B  e.  B )
6 eleq2 2496 . . . . . . 7  |-  ( A  =  B  ->  ( B  e.  A  <->  B  e.  B ) )
76notbid 286 . . . . . 6  |-  ( A  =  B  ->  ( -.  B  e.  A  <->  -.  B  e.  B ) )
85, 7syl5ibr 213 . . . . 5  |-  ( A  =  B  ->  ( Ord  B  ->  -.  B  e.  A ) )
94, 8anim12d 547 . . . 4  |-  ( A  =  B  ->  (
( Ord  A  /\  Ord  B )  ->  ( -.  A  e.  B  /\  -.  B  e.  A
) ) )
109com12 29 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  ->  ( -.  A  e.  B  /\  -.  B  e.  A
) ) )
11 pm4.56 482 . . 3  |-  ( ( -.  A  e.  B  /\  -.  B  e.  A
)  <->  -.  ( A  e.  B  \/  B  e.  A ) )
1210, 11syl6ib 218 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  ->  -.  ( A  e.  B  \/  B  e.  A )
) )
13 ordtri3or 4605 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )
14 df-3or 937 . . . . 5  |-  ( ( A  e.  B  \/  A  =  B  \/  B  e.  A )  <->  ( ( A  e.  B  \/  A  =  B
)  \/  B  e.  A ) )
1513, 14sylib 189 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B )  \/  B  e.  A
) )
16 or32 514 . . . 4  |-  ( ( ( A  e.  B  \/  A  =  B
)  \/  B  e.  A )  <->  ( ( A  e.  B  \/  B  e.  A )  \/  A  =  B
) )
1715, 16sylib 189 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  B  e.  A )  \/  A  =  B
) )
1817ord 367 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( A  e.  B  \/  B  e.  A
)  ->  A  =  B ) )
1912, 18impbid 184 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725   Ord word 4572
This theorem is referenced by:  ordunisuc2  4816  tz7.48lem  6690  oacan  6783  omcan  6804  oecan  6824  omsmo  6889  omopthi  6892  inf3lem6  7580  cantnfp1lem3  7628  infpssrlem5  8179  fin23lem24  8194  isf32lem4  8228  om2uzf1oi  11285  nodenselem4  25631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576
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